Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as engineering, finance, etc. Here, we will discuss the square root of -60.
The square root is the inverse of squaring a number. Since -60 is a negative number, it does not have a real square root. The square root of -60 is expressed in terms of imaginary numbers. In radical form, it is expressed as √-60, which can be simplified to 2√15 * i, where i is the imaginary unit with the property that i² = -1.
The imaginary unit, denoted as i, is used to represent the square roots of negative numbers. It is defined by the property i² = -1. Using the imaginary unit, we can express the square root of any negative number. In practical applications, imaginary numbers are used in complex number calculations which are significant in fields like electrical engineering and control systems.
To find the square root of -60, we first consider the positive part, 60. The prime factorization of 60 is 2 x 2 x 3 x 5. Hence, the square root of 60 can be expressed as √60 = √(2² x 3 x 5) = 2√15. Since the original number is negative, we multiply by i, giving us the square root of -60 as 2√15 * i.
Complex numbers, of which imaginary numbers are a part, are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit. Complex numbers are crucial in various fields, including physics, engineering, and applied mathematics, allowing for the analysis of waveforms, electrical circuits, and more.
The square root of a negative number is always an imaginary number since no real number squared gives a negative result. This property allows us to extend the real number system to include complex numbers, enabling solutions to equations that have no real solutions.
Imaginary numbers often appear in physics and engineering, particularly in solving differential equations and in the analysis of electrical circuits. They help in representing quantities with both magnitude and direction, such as impedance in AC circuits, which is a complex number consisting of resistance (real part) and reactance (imaginary part).
Students often make mistakes when dealing with square roots of negative numbers, such as ignoring the imaginary unit or misapplying the properties of real numbers. Let's explore some common errors and how to avoid them.
Find the square root of -60 in terms of i.
2√15 * i
The square root of -60 is found by separating the negative sign as i.
We first calculate the square root of 60, which is 2√15, and then multiply by i to account for the negative sign: √-60 = √60 * √-1 = 2√15 * i.
What is the square of 2√15 * i?
-60
The square of 2√15 * i is calculated as follows: (2√15 * i)² = (2√15)² * i² = 4 * 15 * (-1) = -60.
If 3i is multiplied by √-60, what is the result?
-90i
To find the result, multiply 3i by √-60, which is 2√15 * i: 3i * 2√15 * i = 6√15 * i² = 6√15 * (-1) = -6√15.
Express the square root of -60 as a complex number.
0 + 2√15 * i
The square root of -60 is purely imaginary, so it can be expressed as the complex number 0 + 2√15 * i, where the real part is 0 and the imaginary part is 2√15.
Simplify the expression i√-60.
-2√15
Simplifying i√-60 involves recognizing that √-60 = 2√15 * i.
Therefore, i * √-60 = i * (2√15 * i) = 2√15 * (i²) = 2√15 * (-1) = -2√15.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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